L(s) = 1 | + (1.24 + 0.664i)2-s + (0.443 + 2.93i)3-s + (1.11 + 1.65i)4-s + (−3.67 − 1.44i)5-s + (−1.39 + 3.96i)6-s + (2.36 − 1.18i)7-s + (0.292 + 2.81i)8-s + (−5.57 + 1.72i)9-s + (−3.63 − 4.24i)10-s + (−1.30 + 4.24i)11-s + (−4.38 + 4.01i)12-s + (−1.69 + 0.387i)13-s + (3.74 + 0.0859i)14-s + (2.61 − 11.4i)15-s + (−1.50 + 3.70i)16-s + (0.673 − 0.459i)17-s + ⋯ |
L(s) = 1 | + (0.882 + 0.469i)2-s + (0.255 + 1.69i)3-s + (0.558 + 0.829i)4-s + (−1.64 − 0.645i)5-s + (−0.571 + 1.61i)6-s + (0.893 − 0.449i)7-s + (0.103 + 0.994i)8-s + (−1.85 + 0.573i)9-s + (−1.14 − 1.34i)10-s + (−0.394 + 1.27i)11-s + (−1.26 + 1.16i)12-s + (−0.471 + 0.107i)13-s + (0.999 + 0.0229i)14-s + (0.674 − 2.95i)15-s + (−0.375 + 0.926i)16-s + (0.163 − 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.460637 + 1.83366i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.460637 + 1.83366i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.24 - 0.664i)T \) |
| 7 | \( 1 + (-2.36 + 1.18i)T \) |
good | 3 | \( 1 + (-0.443 - 2.93i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (3.67 + 1.44i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.30 - 4.24i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (1.69 - 0.387i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.673 + 0.459i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-5.68 + 3.27i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.85 - 2.63i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-2.32 + 4.82i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (0.0831 - 0.144i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.25 + 0.468i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (1.52 + 1.91i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-1.43 - 1.14i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (5.80 - 5.38i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-1.35 - 0.101i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-5.50 + 2.16i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (7.34 - 0.550i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (1.43 + 0.830i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.00 - 0.966i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.69 + 2.50i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.52 - 1.26i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (3.20 - 0.988i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.53700249316631375898038598949, −11.14112444411931055634641806698, −9.867029516971827826802732144545, −8.830266521476033128153307514631, −7.78717501949961231325682034583, −7.36029830661618769377949441354, −5.14693676696622659699488954855, −4.70325491565488377985906536649, −4.14200362378739254527813456777, −3.03923147505544451832890828907,
0.992511108754166746739538710771, 2.67558910048250118579806127205, 3.42827877753785447934738161391, 5.05908291801918250877747568379, 6.20138385530871650627083605269, 7.26878500355889044717217061944, 7.82108746209701757297778030323, 8.635588397623192089512975766023, 10.60848930865925904664612644212, 11.50013945976158093441123878095